On the Frobenius Complexity of Stanley-Reisner Rings Irina Ilioaea - - PowerPoint PPT Presentation

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On the Frobenius Complexity of Stanley-Reisner Rings

Irina Ilioaea

Georgia State University, Atlanta, GA, USA

June, 2020

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Stanley-Reisner Rings and Simplicial Complexes

Non-faces : {x1, x3}, {x1, x4} Facets: {x1, x2}, {x2, x3, x4}

Figure: Simplicial complex ∆

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Stanley-Reisner Rings and Simplicial Complexes

Non-faces : {x1, x3}, {x1, x4} Facets: {x1, x2}, {x2, x3, x4}

Figure: Simplicial complex ∆

The Stanley-Reisner ring associated to our simplicial complex ∆ is given by k[∆] = k[x1, x2, x3, x4] (x1x3, x1x4) .

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Frobenius Operators

Let (R, m, k) a local ring of characteristic p. Let F : R → R be the Frobenius map, that is F(r) = rp. We have that (a + b)p = ap + bp, (a · b)p = ap · bp, for all a, b ∈ R. Therefore, the Frobenius map is a ring homomorphism. Let F e : R → R be the e-th iteration of the Frobenius map, that is F e(r) = rq, where q = pe, e ∈ N.

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Frobenius Operators

Let (R, m, k) a local ring of characteristic p. Let F : R → R be the Frobenius map, that is F(r) = rp. We have that (a + b)p = ap + bp, (a · b)p = ap · bp, for all a, b ∈ R. Therefore, the Frobenius map is a ring homomorphism. Let F e : R → R be the e-th iteration of the Frobenius map, that is F e(r) = rq, where q = pe, e ∈ N. For any e ≥ 0, we let R(e) be the R-algebra defined as follows: as a ring R(e) equals R while the R-algebra structure is defined by rs = rqs, for all r ∈ R, s ∈ R(e). In the same way, starting with an R-module M, we can define a new R-module M(e).

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Frobenius Operators

Let ER := ER(k) denote the injective hull of the residue field k. An e-th Frobenius operator(action) on ER is an additive map φ : ER → ER such that φ(rz) = rqφ(z), for all r ∈ R and z ∈ ER. The collection of e-th Frobenius operators(actions) on ER is an R-module, denoted by Fe(ER).

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Frobenius Operators

Let ER := ER(k) denote the injective hull of the residue field k. An e-th Frobenius operator(action) on ER is an additive map φ : ER → ER such that φ(rz) = rqφ(z), for all r ∈ R and z ∈ ER. The collection of e-th Frobenius operators(actions) on ER is an R-module, denoted by Fe(ER).

Definition (The Frobenius Algebra of Operators)

The algebra of the Frobenius operators on ER is defined by F(ER) =

• e≥0

Fe(ER). This is a N-graded noncommutative ring under composition of maps and due to Matlis duality, its zero degree component is R.

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The Frobenius Algebra of Operators

If (R, m, k) is d-dimensional, local and Gorenstein ring, ER ∼ = Hd

m(R)

and F(Hd

m(R)) is generated by the canonical action F on Hd m(R). In

this case, the Frobenius complexity of the ring R is −∞.

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The Frobenius Algebra of Operators

If (R, m, k) is d-dimensional, local and Gorenstein ring, ER ∼ = Hd

m(R)

and F(Hd

m(R)) is generated by the canonical action F on Hd m(R). In

this case, the Frobenius complexity of the ring R is −∞.

Question(Lyubeznik, Smith - 1999)

Is F(ER) always finitely generated as a ring over R?

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The Frobenius Algebra of Operators

If (R, m, k) is d-dimensional, local and Gorenstein ring, ER ∼ = Hd

m(R)

and F(Hd

m(R)) is generated by the canonical action F on Hd m(R). In

this case, the Frobenius complexity of the ring R is −∞.

Question(Lyubeznik, Smith - 1999)

Is F(ER) always finitely generated as a ring over R? In 2009, Katzman gave an example of a ring R such that F(ER) is not finitely generated as a ring over R. The ring is R = k[x, y, z]/(xy, xz). The Frobenius complexity of the ring R equals 0.

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The Frobenius Algebra of Operators

Katzman raised the finite generation question for the determinantal ring of 2x2 minors in a 2x3 matrix. Enescu and Yao showed that the Frobenius complexity of determinantal rings can be positive, irrational and depends upon the characteristic.

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The Frobenius Algebra of Operators

Katzman raised the finite generation question for the determinantal ring of 2x2 minors in a 2x3 matrix. Enescu and Yao showed that the Frobenius complexity of determinantal rings can be positive, irrational and depends upon the characteristic. In 2012, ` Alvarez, Boix and Zarzuela completely described what happens in the case of Stanley-Reisner rings: when finite generation

• ccurs, then F(ER) is principally generated.

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The Frobenius Algebra of Operators

Problem

Find ways to measure the generation of F(ER) over R.

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The Frobenius Algebra of Operators

Problem

Find ways to measure the generation of F(ER) over R. Enescu and Yao were motivated by the finite generation question to introduce a new invariant of a local ring of prime characteristic, called the Frobenius complexity.

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The Frobenius Algebra of Operators

Problem

Find ways to measure the generation of F(ER) over R. Enescu and Yao were motivated by the finite generation question to introduce a new invariant of a local ring of prime characteristic, called the Frobenius complexity. In the case when this Frobenius algebra is infinitely generated over R, they used the complexity sequence {ce}e≥0 in order to describe how far it is from being finitely generated.

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Frobenius Complexity

Let Ge := Ge(F(ER)) be the subring of F(ER) generated by elements of degree less or equal to e. Note that Ge−1 ⊆ Ge, for all e. Moreover, (Ge)i = F(ER)i, for all 0 ≤ i ≤ e and (Ge−1)e ⊆ F(ER)e. We will denote the minimal number of homogeneous generators of Ge as a subring of F(ER) over F(ER)0 = R by ke.

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Frobenius Complexity

Let Ge := Ge(F(ER)) be the subring of F(ER) generated by elements of degree less or equal to e. Note that Ge−1 ⊆ Ge, for all e. Moreover, (Ge)i = F(ER)i, for all 0 ≤ i ≤ e and (Ge−1)e ⊆ F(ER)e. We will denote the minimal number of homogeneous generators of Ge as a subring of F(ER) over F(ER)0 = R by ke.

Proposition [Enescu, Yao]

The minimal number of generators of the R-module F(ER)e (Ge−1)e equals ke − ke−1, for all e.

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Frobenius Complexity

Definition [Enescu, Yao]

The sequence {ke}e is called the growth sequence for F(ER). The complexity sequence is given by {ce = ke − ke−1}e. The complexity of F(ER) is cx(F(ER)) = inf {n > 0 : ce = O(ne)}.

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Frobenius Complexity

Definition [Enescu, Yao]

The Frobenius complexity of the ring R is defined by cxF(R) = logp(cx(F(ER))). It is easy to note that F(ER) is finitely generated as a ring over R if and only if cx(F(ER))) = 0 if and only if {ce}e≥0 is eventually zero. In this case, cxF(R) = −∞.

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Frobenius Complexity

Definition [Enescu, Yao]

The Frobenius complexity of the ring R is defined by cxF(R) = logp(cx(F(ER))). It is easy to note that F(ER) is finitely generated as a ring over R if and only if cx(F(ER))) = 0 if and only if {ce}e≥0 is eventually zero. In this case, cxF(R) = −∞. If the sequence {ce}e≥0 is bounded by above, but not eventually zero, cx(F(ER))) = 1. Hence, cxF(R) = 0.

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Frobenius Complexity

Let k be a field of characteristic p, S = k[[x1, . . . , xn]] and q = pe, for e ≥ 0. Let I ≤ S be an ideal in S and R = S/I. We denote I [q] = (iq : i ∈ I).

Proposition(Fedder)

There exists an isomorphism of R-modules: Fe(ER) ∼ = I [q] :S I I [q] .

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Frobenius Complexity

For any e ≥ 0 denote Ke := (I [pe] :S I) and Le :=

• 1≤β1,...,βs<e,β1+...+βs=e

Kβ1K [pβ1]

β2

· · · K [pβ1+···+βs−1]

βs

.

Proposition(Katzman)

For any e ≥ 1, let F<e be the R-subalgebra of F(ER) generated by F0(ER), . . . , Fe−1(ER). Then F<e ∩ Fe(ER) = Le. Therefore, (Ge−1)e(F(ER)) ∼ = Le + I [q] I [q] .

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Frobenius Complexity

Let k be a field of characteristic p, S = k[[x1, . . . , xn]] and q = pe, for e ≥ 0. Let I ≤ S be an ideal in S and R = S/I.

Definition [Enescu, Yao]

The complexity sequence {ce}e of the ring R is given by ce = µS

• I [q] :S I

Le + I [q]

• .

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The case of Stanley-Reisner Rings

Let I ≤ S be a square-free monomial ideal in S and R = S/I the Stanley-Reisner ring associated to I.

Theorem (` Alvarez Montaner, Boix and Zarzuela)

The Frobenius algebra F(ER) associated to a Stanley-Reisner ring R is either principally generated or infinitely generated.

Remark

Therefore, for Stanley-Reisner rings, the Frobenius complexity cxF(R) is either −∞ or 0.

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Main Results

Definition

We define Jq to be the unique minimal monomial ideal satisfying the equality (I [q] : I) = I [q] + Jq + (x1)q−1, where (x1)q−1 = xq−1

1

· · · xq−1

n

.

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Main Results

Example

Let I = (x1x5, x2x5, x2x3, x2x4). Then (I [q] : I) = (xq

1 xq 5 , xq 2 xq 5 , xq 2 xq 3 , xq 2 xq 4 , xq−1 1

xq−1

2

xq

5 , xq 2 xq−1 3

xq−1

4

xq−1

5

, xq−1

1

xq−1

2

xq

4 xq−1 5

, xq−1

1

xq−1

2

xq

3 xq−1 5

, xq−1

1

xq−1

2

xq−1

3

xq−1

4

xq−1

5

) and therefore Jq = (xq−1

1

xq−1

2

xq

5 , xq 2 xq−1 3

xq−1

4

xq−1

5

, xq−1

1

xq−1

2

xq

4 xq−1 5

, xq−1

1

xq−1

2

xq

3 xq−1 5

).

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Main Results

Remark

The complexity sequence {ce}e≥0 is bounded by above since ce ≤ µS(Jp) + 1, for any e ≥ 0.

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Main Results

Remark

The complexity sequence {ce}e≥0 is bounded by above since ce ≤ µS(Jp) + 1, for any e ≥ 0.

Remark

In the case when F(ER) is principally generated, it is generated by (x1)p−1. When F(ER) is infinitely generated, F1(ER) has µ + 1 minimal generators, µ of them being the minimal generators of Jp and (x1)p−1. Each graded piece Fe(ER) adds up µ new generators coming from Jq.

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Main Results

Main Theorem(-)

Let k be a field of characteristic p, S = k[[x1, . . . , xn]] and q = pe, for e ≥ 0. Let I ≤ S be a square-free monomial ideal in S and R = S/I its Stanley-Reisner ring. Then, {ce}e≥0 = {0, µ + 1, µ, µ, µ, . . .}, where µ := µS(Jp). Our result generalizes the work of ` Alvarez Montaner, Boix and Zarzuela and settles an open question mentioned by ` Alvarez Montaner in one of his papers.

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Main Results

` Alvarez Montaner defined the generating function of a skew R-algebra using the complexity sequence.

Definition

The generating function of F(ER) is defined as GF(ER)(T) =

• e≥0

ceT e.

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Main Results

Corollary

Let k be a field of characteristic p, S = k[[x1, . . . , xn]] and q = pe, for e ≥ 0. Let I ≤ S be a square-free monomial ideal in S, R = S/I its Stanley-Reisner ring. Then the generating function of the Frobenius algebra of operators is GF(ER)(T) = (µ + 1)T +

• e≥2

µT e = (µ + 1)T − T 2 1 − T .

Proof

Note that c0 = 0. Using the Main Theorem, we have that c1(R) = µ + 1 and ce = µ, for every e ≥ 2.

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Applications of my result

Our theorem describes the complexity sequence of any Stanley-Reisner ring. Moreover, we show that the complexity sequence is independent on the characteristic of the ring in this case.

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Applications of my result

Our theorem describes the complexity sequence of any Stanley-Reisner ring. Moreover, we show that the complexity sequence is independent on the characteristic of the ring in this case. We showed that the Frobenius complexity sequence, which is a positive characteristic invariant of our ring is in fact a combinatorial invariant introduced by ` Alvarez Montaner, Boix and Zarzuela, the number of maximal free pairs of the simplicial complex associated to

• ur ring.

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References

• I. Ilioaea

On the Frobenius Complexity of Stanley-Reisner Rings Communications in Algebra, 2020.

• F. Enescu, Y. Yao

The Frobenius Complexity of a Local Ring of Prime Characteristic Journal of Algebra, 459 (2003), 133 – 156.

• A. Boix, J. Montaner, S. Zarzuela

Frobenius and Cartier Algebras of Stanley-Reisner Rings

• J. Algebra, 358, (2012), 162–177.

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References

• A. Boix, S. Zarzuela

Frobenius and Cartier Algebras of Stanley-Reisner Rings II Acta Mathematica Vietnamica, 324, (2019), 1–16.

• J. `

Alvarez Montaner Generating functions associated to Frobenius algebras Linear Algebra and its Applications, 528, (2019), 310–326.

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Thank you!

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